The actual code of divide and conquer and merge sort

In computer science, divide and conquer is a very important algorithm design paradigm.

It solves the original problem by decomposing a complex problem into several smaller sub-problems, solving these sub-problems separately, and then merging their results.

Merge Sort is a classic example of using divide and conquer.

What is merge sort?.

Merge sort is a comparison-based sorting algorithm that uses a divide-and-conquer strategy for sorting.

It divides the array into two sub-arrays, sorts the two sub-arrays separately, and then merges them into an ordered array.

This process proceeds recursively until each subarray contains only one element.

Merge sort steps.

1. # Decomposition #: Divide the array to be sorted into two sub-arrays from the middle.

2. # Solve #: Recursively merge and sort two subarrays.

3. # Merge #: Combine two sorted subarrays into an ordered array.

Code implementation.

The following is the merge sort code implemented in Python:

def merge_sort(arr):
    """
    归并排序的主函数，接受一个列表作为输入，返回排序后的列表。

    """
    if len(arr) <= 1:
        return arr

    # 找到数组的中间位置
    mid = len(arr) // 2

    # 递归地对左半部分和右半部分进行排序
    left_half = merge_sort(arr[:mid])
    right_half = merge_sort(arr[mid:])

    # 合并两个已排序的部分
    return merge(left_half, right_half)

def merge(left, right):
    """
    合并两个已排序的列表，返回一个新的有序列表。

    """
    sorted_list = []
    left_index, right_index = 0, 0

    # 比较两个列表的元素，按顺序添加到sorted_list中
    while left_index < len(left) and right_index < len(right):
        if left[left_index] < right[right_index]:
            sorted_list.append(left[left_index])
            left_index += 1
        else:
            sorted_list.append(right[right_index])
            right_index += 1

    # 如果左半部分还有剩余元素，添加到sorted_list中
    while left_index < len(left):
        sorted_list.append(left[left_index])
        left_index += 1

    # 如果右半部分还有剩余元素，添加到sorted_list中
    while right_index < len(right):
        sorted_list.append(right[right_index])
        right_index += 1

    return sorted_list

Time complexity analysis.

The time complexity of merge sort can be analyzed by recursive tree.

Assuming that the length of the array is n, each decomposition operation requires O (log n) times, and each merge operation requires O (n) times.

Therefore, the total time complexity is: \[ T(n) = O(n \log n) \] This time complexity is optimal because any comparison-based sorting algorithm requires at least O (n log n) comparisons in the worst case.

Optimization and practical application.

Although the time complexity of merge sort is already optimal, in some practical applications, we may need to consider space complexity and stability.

The space complexity of merge sort is O (n) because it requires additional storage space to hold temporary arrays.

If memory resources are limited, you can consider using In-place Merge Sort (In-place Merge Sort), but this will greatly increase the complexity of the implementation.

In addition, merge sort is a stable sorting algorithm, which means that the relative order of equal elements does not change.

This is very important for some application scenarios, such as when we need to sort multiple fields, we can use merge sort to maintain the original order.

Summarize.

Merge sort is a classic divide and conquer algorithm, which realizes sort by recursively decomposing, solving and merging sub-problems.

Although its implementation is relatively complex, its time complexity is O (n log n), and it is one of the best known sorting algorithms.

In practical applications, merge sorting is widely used because of its stability and suitability for large-scale data.

I hope that through the introduction of this article, you can better understand and apply the powerful tool of merge sorting.